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Analytic aspects of the circulant Hadamard conjecture

Teodor Banica, Ion Nechita, Jean-Marc Schlenker (2014)

Annales mathématiques Blaise Pascal

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for | q 0 | = ... = | q N - 1 | = 1 the quantity Φ = ∑ i + k = j + l q i q k q j q l satisfies Φ ≥ N 2 , with equality if and only if q = ( q i ) is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of Φ , (2) the study of the critical points of Φ , and (3) the computation of the moments of Φ . We explore here these questions,...

Analytical representation of ellipses in the Aitchison geometry and its application

Karel Hron (2009)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Compositional data, multivariate observations that hold only relative information, need a special treatment while performing statistical analysis, with respect to the simplex as their sample space ([Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London, 1986.], [Aitchison, J., Greenacre, M.: Biplots of compositional data. Applied Statistics 51 (2002), 375–392.], [Buccianti, A., Mateu-Figueras, G., Pawlowsky-Glahn, V. (eds): Compositional data analysis in the geosciences:...

Anisotropic complex structure on the pseudo-Euclidean Hurwitz pairs

W. Królikowski (1991)

Annales Polonici Mathematici

The concept of supercomplex structure is introduced in the pseudo-Euclidean Hurwitz pairs and its basic algebraic and geometric properties are described, e.g. a necessary and sufficient condition for the existence of such a structure is found.

Another formulation of the Wick’s theorem. Farewell, pairing?

Igor V. Beloussov (2015)

Special Matrices

The algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory, statistical physics, and quantum chemistry by the standard packages of the well known...

AnS-type upper bound for the largest singular value of nonnegative rectangular tensors

Jianxing Zhao, Caili Sang (2016)

Open Mathematics

An S-type upper bound for the largest singular value of a nonnegative rectangular tensor is given by breaking N = {1, 2, … n} into disjoint subsets S and its complement. It is shown that the new upper bound is smaller than that provided by Yang and Yang (2011). Numerical examples are given to verify the theoretical results.

Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils

Tadeusz Kaczorek (2013)

International Journal of Applied Mathematics and Computer Science

The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-time systems with regular pencils. An equality defining the set of admissible initial conditions for given inputs is derived. The proposed method is illustrated by a numerical example.

Application of the partitioning method to specific Toeplitz matrices

Predrag Stanimirović, Marko Miladinović, Igor Stojanović, Sladjana Miljković (2013)

International Journal of Applied Mathematics and Computer Science

We propose an adaptation of the partitioning method for determination of the Moore-Penrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant...

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